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Mathbox for Scott Fenton |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brtxpsd | Structured version Visualization version GIF version |
Description: Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brtxpsd.1 | ⊢ 𝐴 ∈ V |
brtxpsd.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brtxpsd | ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 4930 | . . 3 ⊢ (𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V))) | |
2 | opex 5213 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | elrn 5665 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉) |
4 | brsymdif 4988 | . . . . . 6 ⊢ (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ¬ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉)) | |
5 | brv 5221 | . . . . . . . . 9 ⊢ 𝑥V𝐴 | |
6 | vex 3418 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
7 | brtxpsd.1 | . . . . . . . . . 10 ⊢ 𝐴 ∈ V | |
8 | brtxpsd.2 | . . . . . . . . . 10 ⊢ 𝐵 ∈ V | |
9 | 6, 7, 8 | brtxp 32868 | . . . . . . . . 9 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ (𝑥V𝐴 ∧ 𝑥 E 𝐵)) |
10 | 5, 9 | mpbiran 696 | . . . . . . . 8 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥 E 𝐵) |
11 | 8 | epeli 5320 | . . . . . . . 8 ⊢ (𝑥 E 𝐵 ↔ 𝑥 ∈ 𝐵) |
12 | 10, 11 | bitri 267 | . . . . . . 7 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥 ∈ 𝐵) |
13 | brv 5221 | . . . . . . . 8 ⊢ 𝑥V𝐵 | |
14 | 6, 7, 8 | brtxp 32868 | . . . . . . . 8 ⊢ (𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉 ↔ (𝑥𝑅𝐴 ∧ 𝑥V𝐵)) |
15 | 13, 14 | mpbiran2 697 | . . . . . . 7 ⊢ (𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉 ↔ 𝑥𝑅𝐴) |
16 | 12, 15 | bibi12i 332 | . . . . . 6 ⊢ ((𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
17 | 4, 16 | xchbinx 326 | . . . . 5 ⊢ (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
18 | 17 | exbii 1810 | . . . 4 ⊢ (∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
19 | 3, 18 | bitri 267 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
20 | exnal 1789 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) | |
21 | 1, 19, 20 | 3bitrri 290 | . 2 ⊢ (¬ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴) ↔ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵) |
22 | 21 | con1bii 349 | 1 ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∀wal 1505 ∃wex 1742 ∈ wcel 2050 Vcvv 3415 △ csymdif 4105 〈cop 4447 class class class wbr 4929 E cep 5316 ran crn 5408 ⊗ ctxp 32818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-symdif 4106 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-eprel 5317 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-fo 6194 df-fv 6196 df-1st 7501 df-2nd 7502 df-txp 32842 |
This theorem is referenced by: brtxpsd2 32883 |
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